Consider a hypothetical atom that has just two states: a

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The partition function equals the sum of all Boltzmann factors, that is:

\(Z=\sum_{s} e^{-E(s) / k T}\)

Consider we have two states, a ground state with zero energy and excited state with energy of \(\epsilon=2 \mathrm{eV}\), therefore the partition function is:

\(\begin{array}{l}
Z=e^{0}+e^{-\epsilon / k T} \\
Z=1+e^{-\epsilon / k T}
\end{array}\)

we need to plot the partition function as a function of temperature T, first substitute with  and with Boltzmann constant in \(\mathrm{eV}\left(k=8.617 \times 10^{-5} \mathrm{eV} / \mathrm{K}\right)\), so:

\(\begin{array}{c}
Z=1+e^{-2 \mathrm{eV} /\left(8.617 \times 10^{-5} \mathrm{eV} / \mathrm{K}\right) T} \\
Z=1+e^{-23210 \mathrm{~K} / T}
\end{array}\)

Using this expression we can plot the partition function as a function of temperature.